Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}x-y &= 4 \\ -7x-8y &= 2\end{align*}$
Solution: Begin by moving the $y$ -term in the second equation to the right side of the equation. $-7x = 8y+2$ Divide both sides by $-7$ to isolate $x$ $x = {-\dfrac{8}{7}y - \dfrac{2}{7}}$ Substitute this expression for $x$ in the first equation. $({-\dfrac{8}{7}y - \dfrac{2}{7}}) - y = 4$ $-\dfrac{8}{7}y - \dfrac{2}{7} - y = 4$ Simplify by combining terms, then solve for $y$ $-\dfrac{15}{7}y - \dfrac{2}{7} = 4$ $-\dfrac{15}{7}y = \dfrac{30}{7}$ $y = -2$ Substitute $-2$ for $y$ in the top equation. $x+ 2 = 4$ $x+2 = 4$ $x = 2$ The solution is $\enspace x = 2, \enspace y = -2$.